91Ó°ÊÓ

What is the degree of the following fields over Q? Prove your assertion. (a) \(Q(\alpha)\) where \(x^{3}=2\) (b) \(Q(x)\) where \(x^{3}=p\) (prime) (c) \(Q(x)\) where \(\alpha\) is a root of \(t^{3}-t-1\) (d) \(Q(x, \beta)\) where \(\alpha\) is a root of \(t^{2}-2\) and \(\beta\) is a root of \(t^{2}-3\) (e) \(Q(\sqrt{-1}, \sqrt{3})\)

Let \(S_{n}\) be the symmetric group of permutations of \(\\{1, \ldots, n\\}\). (a) Show that \(S_{n}\) is generated by the transpositions \((12),(13), \ldots,(1 n) .\) [Hint: Use conjugations. ] (b) Show that \(S_{n}\) is generated by the transpositions \((12),(23),(34), \ldots .(n-1, n) .\) (c) Show that \(S_{w}\) is generated by the cycles \((12)\) and \((123 \cdots n)\). (d) Let \(p\) be a prime number. Let \(i \neq 1\) be an integer with \(1